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Prove that the intersection of all maximal left ideals of a ring $R$ is a two sided ideal.

What i did:Suppose $B$ be the intersection of all maximal left ideals of the ring $R$. Clearly $B$ is a left ideal,so we only have to prove that if $b \in B$, $r \in R$ imply $br \in R$. Equivalently if there is a maximal left ideal $M$ with $br \notin M$, then $ \exists $ a maximal left ideal $A$ with $b \notin A$.Give some idea to show such $A$ will exist.

Arpit Kansal
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  • There is a characterization of this famous ideal ( the Jacobson radical) that makes this fact obvious. – orangeskid Oct 14 '14 at 16:27
  • sorry i don't know about that – Arpit Kansal Oct 14 '14 at 16:31
  • http://en.wikipedia.org/wiki/Jacobson_radical – orangeskid Oct 14 '14 at 16:36
  • @rschwieb Could there be another (and more direct) proof of the fact that the intersection is a two-sided ideals, without characterizing it as intersection of annihilators? My point is, it seems not every answer to the present question would fit under the suggested duplicate. –  Oct 14 '14 at 19:22
  • Dear @carebear : I have never seen a strategy simpler than that one, but let me know if you find one. – rschwieb Oct 15 '14 at 10:32
  • @rschwieb Is is not possible to proceed my idea further somehow? – Arpit Kansal Oct 15 '14 at 13:55
  • @ArpitKansal : I know the feeling: it seems like it should have an elementary answer. My experience, though, is that time spent in this direction for this particular problem usually exceeds the time required to use the alternate method by a lot. – rschwieb Oct 15 '14 at 14:45

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